Problem 1 Probability review Ross, Chapter 3, problem 45:...

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MS&E 221 Problem Set 1 Ramesh Johari Due: January 21, 2016, 5:00 PM , in the basement of Huang Eng. Ctr. Reading. Read Sections 4.1, 4.2, 4.3, and 4.5.1 in Ross. Problem 1 (Probability review). Ross, Chapter 3, problem 45: An individual traveling on the real line is trying to reach the origin. However, the larger the step desired, the greater is the variance in the result of that step. Specifically, whenever the person is at location x , he next moves to a location having mean 0 and variance βx 2 . Let X n denote the position of the individual after having taken n steps. Supposing that X 0 = x 0 , find (a) E [ X n ] (b) V ar ( X n ) Solution:
Problem 2. For each of the following, explain whether it is reasonable to model the given process as a Markov chain. (a) The daily closing value of the S&P 500. (b) The sequence of bids observed on a single eBay auction. (c) The times at which patients arrive to the emergency room at a hospital.
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S t ( μ t + Z t ) , where μ t are constants and Z t are independent. No, in many cases, historical prices of stocks are used in predicting the future prices. For ex- ample, this is the basis for momentum based strategies. (b) Yes, when a new bidder arrives, the current bid is the highest among all previous bids. One would naturally assume that the new bid is the current bid plus an increment that is independent of the bidding history. No, sometimes a new bidder would look at a couple of recent bids to determine his bid. If recent bids demonstrate a sharp increase, the new bidder should consider a more competitive bid than usual as the product is probably more popular amongst customers. (c) Yes, it is natural to assume that the time intervals between two consecutive patient arrivals are independent. For example, this is the reasoning behind the Poisson process that will be studied later in the course. Thus the arriving time process is a Markov chain. No, the arrivals could actually be correlated, for example, when there is a disaster or a major accident there will be a burst of arrivals within a short span of time.

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